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We usually follow the definitions and notation of Hartshorne (1977) and Kollár and Mori (1998). Those that may be less familiar or are used inconsistently in the literature are recalled in Section 1.1.

The rest of the chapter is more advanced. We suggest skipping it at first reading and then returning to these topics when they are used later.

The classical theory of minimal models is summarized in Section 1.2. Minimal and canonical models of pairs are treated in greater detail in Section 1.3. Our basic reference is Kollár and Mori (1998), but several of the results that we discuss were not yet available when Kollár and Mori (1998) appeared. In Section 1.4 we collect various theorems that can be used to improve the singularities ...

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